1. Sketch the parametric curve x = cost, y = cos(2t) (0 < t ≤ 2π), showing its direction.
Find a Cartesian Equation in x and y whose graph contains the parametric curve.
2. Find the coordinates of the points at which the parametric curve x = t
, y = t
3 − 3t
a) a horizontal tangent
b) a vertical tangent.
3. Find the Cartesian equation of the tangent to the parametric curve x = 3t + t
, y =
3 − 9t
2 at the point t = 1.
4. For the curve x = t − sin t, y = 1 − cost, nd d
, and determine where the curve is
concave up and concave down.
5. Find the length of the curve x = t − sin t, y = 1 − cost, 0 ≤ t ≤ 2π. Hint: to
evaluate the integral use the identity sin2
6. Find the area of the surface generated by rotating the curve x = 3t − t
, y = 3t
, 0 ≤
t ≤ 1 about the x-axis.
7. Transform the polar equation r =
3−4 sin θ
to rectangular coordinates.
8. Find the area of the region that lies inside the graphs of both polar equations r = 1
and r = 2 sin θ.
9. Find the length of the polar curve r = e
, 0 ≤ θ ≤ 3π.
10. Find the limit of the sequence an =
, if it exists.
11. Find the sum of the series P∞
12. Calculate the sum of the series:
n(n + 2)
13. Determine whether each of the following series converge or not. (Name the test you
use. You do not have to evaluate the sums of those series.)
n ln n
14. Determine whether each of the following series converge absolutely, converge conditionally or diverge.
15. Starting with the geometric series 1 + x − x
2 + x
3 + . . . (−1 < x < 1), nd
the power series for x
in powers of x.
a) Where is the series valid ?
b) Using the result in (a) nd the sum P∞
16. Determine the radius and interval of convergence of the series P∞
17. Find the Maclaurin series for f(x) = x
starting with a familiar series. For
that values of x is the representation valid ?
18. Find the Taylor series representation of f(x) = ln x in powers of x − 1. Find the
radius of convergence of this Taylor series.
These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction
of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice.
Unethical use is strictly forbidden.